Theorem

Let {Xt}t∈T\{X_t\}_{t \in T} be a Gaussian process, and let T0⊆T1⊆⋯⊆TT_0 \subseteq T_1 \subseteq \dots \subseteq T be a sequence of subsets such that |T0|=1\lvert T_0 \rvert = 1 and |Tn|≤22n\lvert T_n \rvert \leq 2^{2^n} for n≥1n \geq 1. Then,

𝔼[supt∈T⁡Xt]≤O(1)supt∈T⁡∑n≥02n/2d(t,Tn)\mathbb{E}[\sup_{t \in T} X_t] \leq O(1) \sup_{t \in T} \sum_{n \geq 0} 2^{n/2} d(t,T_n)

where for s,t∈Ts,t \in T, the canonical distance is defined as d(s,t)=𝔼|Xs−Xt|2d(s,t) = \sqrt{\mathbb{E}\lvert X_s - X_t \rvert^2}

References

  1. https://tcsmath.wordpress.com/2010/07/18/the-majorizing-measures-theorem/
  2. https://homes.cs.washington.edu/~jrl/cse599wi23/notes/mm2.pdf
  3. Michel Talagrand. "Majorizing measures: the generic chaining." Ann. Probab. 24 (3) 1049 - 1103, July 1996. https://doi.org/10.1214/aop/1065725175